The topic of this book is the physics of transition metal compounds. In all their properties strong electron correlations play a crucial role. However TM compounds are not the only materials in which electron correlations are extremely important. Other such systems are substances containing rare earth elements with partially filled 4 f shells or actinide compounds with 5 f -electrons. These systems show a lot of very interesting special properties such as mixed valence and heavy fermion behavior. And though these phenomena were discovered and are mostly studied in 4 f and 5 f systems, similar effects, maybe less pronounced, are also observed in some TM compounds. The main concepts, and also the main problems in the physics of rare earth (and actinide) compounds are very similar to those in TM systems. Therefore we also include in this book, formally devoted to TM materials, this short chapter in which we summarize the main phenomena discovered in 4 f and 5 f systems, and compare them and their description with that of TM systems. Some of the sephenomena were even discovered first in materials with TM ions, but later proved to be essential in treating rare earth systems; while other notions were introduced for rare earth compounds and later transferred to the study of transition metal systems.

There exists a significant body of literature devoted specifically to some of the topics discussed briefly below. One can find detailed descriptions for example in Hewson (1993) or Coleman (2007).

Transition metal (TM) compounds present a unique class of solids. The physics of these materials is extremely rich. There are among them good metals and strong, large-gap insulators, and also systems with metal–insulator transitions. Their magnetic properties are also very diverse; actually, most strong magnets are transition metal (or rare earth) compounds. They display a lot of interesting phenomena, such as multiferroicity or colossal magnetoresistance. Last but not least, high-Tc superconductors also belong to this class.

Transition metal compounds are manifestly the main area of interest and the basis for a large field of physical phenomena: the physics of systems with strong electron correlations. Many novel ideas, such as Mott insulators, were first suggested and developed in application to transition metal compounds.

From a practical point of view, the magnetic properties of these materials have been considered and used for a long time, but more recently their electronic behavior came to the forefront. The ideas of spintronics, magnetoelectricity and multiferroicity, and high-Tc superconductivity form a very rich and fruitful field of research, promising (and already having) important applications.

There are many aspects of the physics of transition metal compounds. Some of these are of a fundamental nature – the very description of their electronic structure is different from the standard approach based on the conventional band theory and applicable to standard metals such as Na or Al, or insulators or semiconductors such as Ge or Si.

As discussed at the beginning of Chapter 7, there can exist different types of ordering phenomena connected with charge degrees of freedom. Examples are ordering of charges themselves (charge “monopoles”); ordering of electric dipoles, giving ferroelectricity (FE); or ordering of electric quadrupoles, which happens in orbital ordering. We have already discussed the first and third possibilities; we now turn to the second.

Ferroelectricity is a broad phenomenon, in no way restricted to TM compounds. There are ferroelectrics among organic compounds, in some molecular crystals, in systems with hydrogen bonds. But the best, and most important in practice, are ferroelectrics on the basis of TM compounds such as the famous BaTiO3, or the widely used Pb(ZrTi)O3 (“PZT”). And it is in these compounds that one also sometimes meets a very interesting interplay of ferroelectricity and magnetism – the field now known mostly as multiferroicity. By multiferroics (Schmid, 1994) we refer to materials which are simultaneously ferroelectric and magnetic – possibly ferro- or ferrimagnetic, although such cases are rather rare, most of the known multiferroics being antiferromagnetic. (Sometimes ferroelastic systems are alsoincluded in this class.) In this chapter we discuss these classes of compounds, paying attention mostly to the microscopic mechanisms of ferroelectricity and its eventual coupling to magnetism.

A general treatment of ferroelectricity, dealing mainly with the macroscopic aspects of ferroelectrics and their phenomenological description, with special attention paid to practical applications, may be found in many books, for example Megaw (1957), Lines and Glass (1977), Scott (2000), Gonzalo (2006), Blinc (2011).

Crystal field splitting

When we put a transition metal ion in a crystal, the systematics of the corresponding electron states changes. For isolated atoms or ions we have spherical symmetry, and the corresponding states are characterized by the principal quantum number n, by orbital moment l and, with spin–orbit coupling included, by the total angular momentum J. When the atom or ion is in a crystal, the spherical symmetry is violated; the resulting symmetry is the local (point) symmetry determined by the structure of the crystal. Thus, if a transition metal ion is surrounded by a regular octahedron of anions such as O2− (Fig. 3.1) (this is a typical situation in many TM compounds, e.g. in oxides such as NiO or LaMnO3), the d levels which were fivefold degenerate in the isolated ion (l = 2; lz = 2, 1, 0, −1, −2) are split into a lower triplet, t2g, and an upper doublet, eg (Fig. 3.2). The corresponding splitting is caused by the interaction of d-electrons with the surrounding ions in the crystal, and is called crystal field (CF) splitting. The type of splitting and the character of the corresponding levels is determined by the corresponding symmetry. The detailed study of such splittings is a major field in itself, and is mostly treated using group-theoretical methods.

The main topic of this book is the physics of solids containing transition elements: 3d − Ti, V, Cr, Mn, … 4d − Nb, Ru, … 5d −Ta, Ir, Pt, … These materials show extremely diverse properties. There are among them metals and insulators; some show metal–insulator transitions, sometimes with a jump of conductivity by many orders of magnitude. Many of these materials are magnetic: practically all strong magnets belong to this class (or contain rare earth ions, the physics of which is in many respects similar to that of transition metal compounds). And last but not least, superconductors with the highest critical temperature also belong to this group (high-Tc cuprates, with Tc reaching ∼ 150 K, or the recently disovered iron-based (e.g., FeAs-type) superconductors with critical temperature reaching 50–60 K).

The main factor determining the diversity of behavior of these materials is the fact that their electrons may have two conceptually quite different states: they may be either localized at corresponding ions or delocalized, itinerant, similar to those in simple metals such as Na (and, of course, their state may be something in between). When dealing with localized electrons, we have to use all the notions of atomic physics, and for itinerant electrons the conventional band theory may be a good starting point.

Until now we have largely been discussing the properties of correlated systems with integer number of electrons; only in a few places, for example in the sections on charge ordering and on the double exchange, did we touch on some properties of doped correlated systems. But in principle the variety of phenomena which can occur in such systems with the change in electron concentration is quite broad – from a strong modification of magnetic properties up to a possibility of obtaining non-trivial, possibly high-temperature superconducting states.

A number of questions arises when we start thinking about doped strongly correlated systems. Would the system be metallic? And if so, would it be a normal metal described by the standard Fermi liquid theory? In effect, even with partially filled bands the electron correlations can still remain strong, with the Hubbard's U (much) bigger than the bandwidth; thus these questions are really nontrivial.

The other question is, what magnetic properties will result when we dope Mott insulators? As we have argued in Chapter 1 and Section 5.2, for partially filled bands the chances of ferromagnetic ordering are strongly enhanced, whereas Mott insulators with integeroccupation of d-shells are typically antiferromagnetic.

One may also expect that some other, new features could appear in strongly correlated systems with partial occupation of d levels.

When dealing with transition metal compounds one has to look at the different degrees of freedom involved and their interplay. These degrees of freedom are charge, spin, and orbitals. And of course all electronic phenomena occur on the background of the lattice, that is one always has to think about the role of the interaction with the lattice, or with phonons.

The electron spins are responsible for different types of magnetic ordering. The orbitals, especially in the case of orbital (or Jahn—Teller) degeneracy, also lead to a particular type of ordering, and the type of orbital occupation largely determines the character of magnetic exchange and of the resulting magnetic structures.

As to charges, the first question to ask is whether the electrons have to be treated as localized or itinerant. We actually started this book by discussing two possible cases: a band description of electrons in solids, in which the electrons are treated as delocalized, and the picture of Mott insulators, with localized electrons.

But even for localized electrons there still exists some freedom, which has to do with charges. In some systems charges may be disordered in one state, for example at high temperatures, and become ordered at low temperatures. This charge ordering (CO) will be the main topic of this chapter. But, to put it in perspective, we will start by discussing different possible types of ordering, connected with charge degrees of freedom.

The history of the development of some of the key concepts discussed in this book is quite interesting and has some rather unexpected twists and turns. In this section we discuss briefly the history of the concepts of Mott insulators, the Jahn–Teller effect, and the Peierls transition.

Mott insulators and Mott transitions

The notion of a Mott insulator as a state conceptually different from the standard band-like insulators and metals can be introduced using two approaches. In the main text, for example in Chapter 1 we described the approach that uses the Hubbard model (1.6) with short-range (on-site) electron-electron repulsion and attributes the insulating nature for strong interaction to the fact that an electron transferred to an already occupied site experiences repulsion from the electron already sitting on that site. This is the picture most often used nowadays to explain the idea of Mott insulators.

But historically these ideas first appeared in a different picture, presented in a paper by Mott published in 1949 (Mott, 1949) – although it already contained some hints about the picture mostly used nowadays, formalized in the Hubbard model. But the main arguments of Mott in this paper rely rather on the long-range character of Coulomb interaction, and the main statement is that, starting from an insulator, one cannot get a metal by exciting as mall number of electrons and holes.

General theory

In several places in this book we have used the language and notions first developed by Landau to describe second-order phase transition, but which are used nowadays in a much broader context. Here we summarize the basics of this theory and illustrate different situations in which it is used. One can find a more detailed description for example in the brilliant original presentation of Landau and Lifshitz (1969), or in Khomskii (2010) (which is more or less followed below).

The original aim of Landau was to describe II order phase transitions – transitions in which a certain ordering, for example ferromagnetic, appears with decreasing temperature at some critical temperature Tc in a continuous manner. But it turned out later that the approach developed has much broader applicability than originally planned.

In thermodynamics and in statistical physics the optimal equilibrium state of a many-particle system is determined by the condition of the minimum of the Helmholtz freeenergy

F(V, T) = E − TS

or of the Gibbs free energy

Φ(P, T) = E − TS + PV

at given temperature and either fixed volume (C.1) or fixed pressure (C.2); more often in reality we are dealing with the second situation. When a certain ordering appears in the system – it may be magnetic ordering, for example ferro- or antiferromagnetic; or ferroelectricity; or an ordering in a structural phase transition – one can introduce a measure of such ordering, different for specific situations, which is called the order parameter; let us denote it η.

Part IV presents advanced techniques and methods that are useful for understanding phase transitions in materials. The emphasis is on aspects of free energy, energy, entropy, and kinetic processes, and less on specific phase transformations. The chapters are far from a complete set of advanced topics, however, and other topics can be argued to be just as important. The topics in Part IV have proved their value, though, and appear in the literature with some frequency. The reader is warned that some of the presentations assume a higher level of mathematics or physics than the other sections in the book, and some important results are stated without proof.

The chapters in Part IV follow no natural sequence, and may be selected for interest or need. Some topics on energy (Chapter 21), entropy (Chapter 24), and atom movements (Chapter 23) are continuations of content in Chapters 6, 7, 9 of Part II. Chapter 19 presents analyses of phase boundaries at low and high temperatures, and Chapter 20 presents techniques for analyzing thermodynamics and physical properties very close to a critical temperature.